Question

A manufacturer claims that the mean lifetime, u, of its light bulbs is 46 months. The standard deviation of these lifetimes is 8 months. One hundred fifty bulbs are selected at random, and their mean lifetime is found to be 48 months. Can we conclude, at the 0.05 level of significance, that the mean lifetime of light bulbs made by this manufacturer differs from 46 months?

Answer 1

To determine if the mean lifetime of light bulbs made by this manufacturer differs from 46 months, we can conduct a hypothesis test. We reject the null hypothesis and conclude that the mean lifetime differs from 46 months.

Step-by-step explanation:

To determine if the mean lifetime of light bulbs made by this manufacturer differs from 46 months, we can conduct a hypothesis test.

First, we state the null hypothesis, which is that the mean lifetime of the light bulbs is 46 months. The alternative hypothesis is that the mean lifetime is not 46 months.

Next, we calculate the test statistic using the formula:

t = (sample mean - population mean) / (standard deviation / sqrt(sample size))

In this case, the sample mean is 48 months, the population mean is 46 months, the standard deviation is 8 months, and the sample size is 150. Plugging in these values, we find the test statistic to be approximately 2.5.

Finally, we compare the test statistic to the critical value from the t-distribution at a significance level of 0.05. If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the mean lifetime differs from 46 months. Otherwise, we fail to reject the null hypothesis.

In this case, the critical value for a two-tailed test with a significance level of 0.05 is approximately ±1.96. Since the test statistic of 2.5 is greater than 1.96, we reject the null hypothesis and conclude that the mean lifetime of light bulbs made by this manufacturer differs from 46 months.